Practice question · Multiple choice
The span of two vectors in ordinary space is usually a plane, but sometimes only a line. Why does the number of vectors fail to determine the dimension of what they sweep out?
Hints
- Take v and 2v. What does adding 2v to the span of v get you?
- Ask what a vector must do to enlarge the span.
Show the answer
C. Because span counts independent directions, not vectors.
Why
Counting vectors counts the list; dimension counts the independent directions in it. Every combination of (1,2,1) and (2,4,2) lies on the line the first already swept, the second was free and useless. Which is why dimension is defined via a basis, spanning and independent, rather than by counting whatever you happened to write down.
Practise Span and Linear Combinations
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More questions on Span and Linear Combinations
- Adding a vector to a list always enlarges its span.
- Let W be the span of (1, 0, 1) and (0, 1, 1). Decide for each vector whether it lies in W.
- The span of a set is always a subspace, whatever vectors you start from. Why is that automatic?
- A linear system w = c1 v1 + … + ck vk has no solution for the scalars ci. What does this algebraic failure…
- Order the steps that decide whether a vector w lies in the span of v1 and v2.